In disordered systems, the presence of random interactions among the constituting units (see e.g., [1]}) implies that the coupling matrix is a random variable, with the crucial problem being the computation of the quenched free-energy, i.e. the expectation of the free-energy \(F_{N, \beta , J}:=-\frac{1}{\beta } \log Z_{N,\beta , J}\)  w.r.t. the probability distribution of the weights \(J\) . This is motivated by the existence of self-averaging theorems [2]}, [3]} ensuring that the fluctuations of the free-energy w.r.t. its expectation value vanish in the thermodynamic limit \(N\rightarrow \infty \) .
In a nutshell, the quenched free-energy paints a physical situation where we study the thermalization of the neurons at fixed (but random) weights \(J_{ij}\) , thus averaging over all their possible realizations after taking the logarithm of the partition function: this procedure intrinsically matches the so-called adiabatic hypothesis, prescribing that the coupling evolution, if any, takes place on much longer time-scales w.r.t. the neural dynamics, such that couplings can be considered as fixed (or quenched) when addressing the neural dynamics.
